On a boundary value problem for a system of differential equations modeling the electrical activity of the brain

A. S. Patrina

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 144, pp. 383-394

Résumé

The Hopfield-type model of the dynamics of the electrical activity of the brain which is a system of differential equations of the form \begin{equation*} \dot{v}_{i}= -\alpha v_{i}+\sum_{j=1}^{n}w_{ji}f_{\delta}(v_{j})+I_{i}( t), \quad i=\overline{1,n}, \quad t\geq 0, \end{equation*} is under discussion. The model parameters are assumed to be given: $\alpha>0,$ $\tau_{ii}=0,$ $w_{ii}= 0,$ $\tau_{ji}\geq 0$ and $w_{ji}>0$ at $i\neq j,$ $I_{i}(t)\geq 0$ at $t\geq 0.$ Activation function $f_{\delta}$ ($\delta$ — the time of the transition of a neuron to the state of activity) is considered of two types: $$ \delta=0 \ \Rightarrow f_{0}(v)=\left\{ \begin{array}{ll} 0, &v\leq\theta,\\ 1, &v>\theta; \end{array}\right. \ \ \ \ \ \ \delta> 0 \ \Rightarrow \ f_{\delta}(v)=\left\{ \begin{array}{ll} 0, & v\leq \theta,\\ {\delta}^{-1}( v-\theta), & \theta < v \leq \theta+\delta,\\ 1, &v>\theta+\delta. \end{array}\right.$$ For the system of differential equations under consideration, a boundary value problem with the conditions ${v_{i}(0)-v_{i}(T)=\gamma_{i},}$ $i=\overline{1,n},$ is studied. In both cases $\delta= 0$ (discontinuous function $f_{0}$) and $\delta > 0$ ($f_{0}$ continuous function), a solution exists, and if $${\delta} > \frac{T|W|_{\mathbb{R}^{n}\to \mathbb{R}^{n}}}{1 - e^{-\alpha T}}, \quad \text{where} \quad W=(w_{ij})_{n\times n}, $$ the problem has a unique solution. The work also provides estimates for the solution and its derivative. Theorems on fixed points of continuous mappings of metric and normed spaces and on fixed points of monotonic mappings of partially ordered spaces are used. The results obtained are applied to the study of periodic solutions of the differential system under consideration.

Détail

Keywords: neural network, differential equation with a discontinuous right-hand side, boundary value problem, Green's function, mappings of partially ordered spaces, periodic solution
Mots-clés : existence of a solution

Bibliographie

[1] J. J. Hopfield, “Neural networks and physical systems with emergent collective computational properties”, Proc. Nat. Acad. Sci., 79:8 (1982), 2554–2558 | DOI | MR | Zbl

[2] V. L. Bykov, Cytology and General Histology, Sothis Publ., St. Petersburg, 2018, 237 pp. (In Russian)

[3] P. Van den Driesche, X. Zou, “Global attractivity in delayed Hopfield neural network models”, SIAM J. Appl. Math., 58 (1998), 1878–1890 | DOI | MR | Zbl

[4] A. S. Lanina, E. A. Pluzhnikova, “On the properties of solutions to differential systems modeling the electrical activity of the brain”, Vestnik rossiyskikh universitetov. Matematika = Russian Universities Reports. Mathematics, 27:139 (2022), 270–283 (In Russian) | Zbl

[5] E. S. Zhukovskii, “Neravenstva Volterra v funktsionalnykh prostranstvakh”, Matem. sb., 195:9 (2004), 3–18 | DOI | MR | Zbl

[6] E. S. Zhukovskii, “Ob uporyadochenno nakryvayuschikh otobrazheniyakh i neyavnykh differentsialnykh neravenstvakh”, Differentsialnye uravneniya, 52:12 (2016), 1610–1627 | DOI | MR | Zbl

[7] E. O. Burlakov, E. S. Zhukovskiy, “On absrtact Volterra equations in partially ordered spaces and their applications”, CONCORD-90: Mathematical Analysis With Applications, International conference in honor of the 90th Birthday of Constantin Corduneanu (2018, Ekaterinburg, Russia), v. 318, Springer Proceedings in Mathematics Statistics, eds. S. Pinelas, A. Kim, V. Vlasov, 2020, 3–11 | DOI | MR | Zbl

[8] S. Benarab, Z. T. Zhukovskaya, E. S. Zhukovskii, S. E. Zhukovskii, “O funktsionalnykh i differentsialnykh neravenstvakh i ikh prilozheniyakh k zadacham upravleniya”, Differentsialnye uravneniya, 56:11 (2020), 1471–1482 | DOI | MR

[9] L. V. Kantorovich, G. P. Akilov, Functional Analysis, Nauka Publ., Mscow, 1984 (In Russian) | MR

[10] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, 5-e izd., Fizmatlit, M., 2019 ; A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, v. I, II, Dover Publications, Mineola, New York, 1957, 1961 | MR | MR

[11] A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy, “Coincidence points principle for mappings in partially ordered spaces”, Topology and its Applications, 179:1 (2015), 13–33 | DOI | MR | Zbl

[12] L. A. Lyusternik, V. I. Sobolev, A Short Course in Functional Analysis, Higher School Publ., Moscow, 1982, 271 pp. (In Russian) | MR

[13] N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations, Nauka Publ., Mscow, 1991 (In Russian) | MR